The midpoint of start overline, AB, end overline 
AB
  is M, left parenthesis, 4, comma, minus, 7, right parenthesisM(4,−7). If the coordinates of AA are left parenthesis, 7, comma, minus, 6, right parenthesis(7,−6), what are the coordinates of BB?

To find the coordinates of \( B \), we can use the midpoint formula. The formula for the midpoint \( M \) of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is:

\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

We are given:

- Midpoint \( M(4, -7) \)
- Point \( A(7, -6) \)

Let the coordinates of point \( B \) be \( B(x_2, y_2) \). Using the midpoint formula, we can set up the following equations:

For the x-coordinate:
\[
\frac{7 + x_2}{2} = 4
\]

For the y-coordinate:
\[
\frac{-6 + y_2}{2} = -7
\]

Now, solve for \( x_2 \) and \( y_2 \).

### Solving for \( x_2 \):
\[
\frac{7 + x_2}{2} = 4
\]
Multiply both sides by 2:
\[
7 + x_2 = 8
\]
Subtract 7 from both sides:
\[
x_2 = 1
\]

### Solving for \( y_2 \):
\[
\frac{-6 + y_2}{2} = -7
\]
Multiply both sides by 2:
\[
-6 + y_2 = -14
\]
Add 6 to both sides:
\[
y_2 = -8
\]

Thus, the coordinates of point \( B \) are \( B(1, -8) \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you find the length of line segment \( AB \)?
2. What is the equation of the line passing through points \( A(7, -6) \) and \( B(1, -8) \)?
3. How does the distance formula relate to the midpoint formula?
4. Can the midpoint formula be used in three-dimensional space? How?
5. What happens to the midpoint when both points are located on the same axis?

**Tip**: When using the midpoint formula, remember it's always the average of the x-coordinates and the average of the y-coordinates of the endpoints.

The midpoint of segment AB is M(4,−7). If the coordinates of A are (7,−6), what are the coordinates of B?

This problem involves using the midpoint formula to find the coordinates of point B given the midpoint M(4,-7) and point A(7,-6). Learn the step-by-step process of solving for x2 and y2, suitable for students in Grades 8-10.

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